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1 Practice A car starts from rest and travels upwards along a straight road inclined at an angle of 5 from the horizontal. The length of the road is 450 m and the mass of the car is 800 kg. The speed of the car increases at a constant rate and is 28 m/s at the top of the slope. Determine the acceleration of the car and the time it takes for the car to reach the top of the slope. Newton s 3 Laws of Motion 1st Law of Inertia (an object at rest remains at rest, an object in motion remains in motion UNLESS acted on by an outside force). 2nd Force/Acceleration Relationship (the acceleration created by forces acting on a mass is equal to the vector sum of the forces divided by the mass of the object). 3rd Action/Reaction Pairs (If object A exerts a force on object B, object B is exerting an equal but opposite force on A). Recall Forces a push or pull acting on an object; a vector quantity measured in Newtons (kg m/s²) Any object experiencing acceleration (a non-constant velocity in either speed or direction) is experiencing out-of-balance forces.

2 Definitions Vector Measurements that have both MAGNITUDE (amount) and DIRECTION; such as displacement, velocity, acceleration and force. Net Force the vector sum of forces acting on an object Equilibrium the condition when the NET FORCE (vector sum) is 0; condition of a = 0 m/s²; stationary objects or objects under constant velocity Definitions Equilibrium Equations Equations, broken down by axis, showing the sum of all forces equal to zero. Non-Equilibrium Equation Equations, broken down by axis, showing the sum of all forces equal to mass x acceleration. Centripetal Force ANY force causing an object to move in circular motion is a centripetal force and is equal to F c = mv² / r Where m = mass, v = tangential velocity, and r = radius of curvature

3 Types of Force Force Symbol Definition Direction Friction F f Contact force that opposes sliding motion between surfaces Normal F N The contact force exerted by a surface on an object Parallel to surface and opposite direction of travel Perpendicular to and away from the surface Spring F K A restoring force that pulls on an object Opposite displacement Tension F T The pull exerted by a rope or string Away from the object and parallel to the string Thrust F thrust A generic term for forces that move objects such as rockets, etc Weight F W A field force due to gravitational attraction between masses Drag F drag A force acting against motion exerted by objects moving through gas or fluids. In direction of acceleration of object Straight down towards center of exerting mass Against motion OR in the direction of fluid flow. Drag Force and Terminal Velocity In most problems, we ignore the effect of air resistance. In these problems, we assume that falling objects continue to accelerate until they hit the ground. In reality, falling objects experience a force due to air resistance drag force. This same force acts on objects being pulled through liquids. As the velocity of an object increases, the drag force increases. Terminal velocity occurs when the drag force equals the force due to gravity. Practice AS/A2 Force Problems

4 Easy Vector Problem Two forces of magnitude 6.0N and 8.0N act at a point P. Both forces act away from point P and the angle between them is 40. Determine the magnitude and direction of the resultant force acting on point P. P 40 Tension Tension is the specific name given to the force exerted by a string, rope or wire. In most physics problems, we assume that these strings are massless. Examine the simple system of a 100 kg mass on a rope. The forces acting on the mass include the gravitational attraction between the Earth and the mass and the tension between the mass and the string. Tension Problems An Atwood machine uses a counterweight to raise and lower an elevator. If the 850-kg elevator is attached via a pulley to a 1000-kg counterweight, what is the acceleration of the system and force of tension in the cable? How does this change when the elevator is full and has a new mass of 1150-kg?

5 Normal Force Any object not accelerating downward at a rate of -9.8 m/s 2, must be experiencing at least one force that is counteracting the pull of gravity This force is frequently the normal force exerted by surface contact and acting perpendicular to the surface. Find acceleration & final velocity Elevators and Scales The scale shows the normal force NOT your weight. When acceleration 0 m/s²; ΣF = ma = F W + F N = -F W + F N The scale will show F N = ma + F W

6 Lab Help Drag Force Lab: In the final question, they said that F drag = bv n At terminal velocity: F drag = F g So mg = bv n Take the log of each side log(m) + log(g) = log(b) + log(v n ) log(m) + log(g) = log(b) + nlog(v) Rearrange to put your DEPENDENT (v) on one side log(v) = [1/n][log(m)] + [1/n][log(b) - log(g)] From this, how does the slope of your line relate to n and b? How does the y-intercept relate to n or b? Frictional Force There are two types of friction force both always oppose movement. When objects are moving they experience kinetic friction. When objects are stationary, they experience static friction. See page 90 in Giancoli for information of the coefficients of friction for various substances

7 Friction and Normal Forces There is also a relationship between the normal force (the force created by two surfaces in contact) and the friction force. As normal force increases, so does friction force. The two are related as follows: F f kinetic = kf N F f static = sf N Find acceleration & final velocity Incline Problems In evaluating a problem where forces act at angles other than 90 and 180 degrees requires establishing a coordinate system and breaking force vectors into component vectors. Inclined planes are one of the most common examples.

8 Incline Coordinate Systems Breaking forces into component vectors: Set-Up the Problem A box is resting on an inclined plane with an angle of 25 degrees from the horizontal. The mass of the box is 10 kg. If the box is at rest, what is the coefficient of friction between the surface of the box and the incline plane? Example 2 A 10-kg box is resting on a 15 incline with a coefficient of friction of μ = A rope connects the 10-kg box to 3-kg box via a pulley. Consider the ropes to be massless and the pulley to be frictionless. Is the system in equilibrium?

9 Working With Angles Always resolve a problem in to a set of perpendicular planes Horizontal and Vertical Parallel and Perpendicular Break vectors into their components Look at equilibrium/non-equilibrium in both axes. Put the final answer back together. Center of Mass The center of mass (sometimes called the center of gravity) is the average position of all the particles of mass that make up an object. In our previous problems, we have assumed that the geometric center of the object is the center of the mass. Center of Gravity In strong gravitational fields, especially for large objects, we can t assume gravity is uniform. In these cases, the center of gravity is located below the center of mass.

10 Torque The MOMENT of the force Force is needed to initiate or stop motion Torque is needed to initiate or stop rotation. Mathematically, torque is calculated by: = F r Torque is often referred to as MOMENT. Couples Two complimentary torques (moments) create a COUPLE. Opposing torques are not a couple. Torque Torque occurs in circular motion and is created by the force PERPENDICULAR to the radius of the movement. If force is not applied at 90, only the part of the force acting perpendicular to the radius creates torque.

11 Practice 3) Compute the magnitude F for each of the two forces acting on the outer circle, so that they create the same couple moment as that created by the 100-N forces acting on the inner circle. Rotational Inertia Objects moving in circular motion tend to remain in circular motion or more simply, spinning objects tend to keep spinning. For rotational inertia: the further the bulk of the mass is to the axis of rotation, the greater the rotational inertia. This can be written mathematically as: I = mr² So, larger masses have more inertia or they are harder to get started and harder to stop AND longer radii have more inertia. Moments of Inertia

13 Choosing an Axis of rotation In any problem of torque, you must select a point as the axis of rotation. In many problems this point will be a pivot, hinge, etc. Sometimes you may wish to select a point of rotation so that it eliminates any unknown torques acting at that point (radius = 0m) Problem 1 Ch 9 #8: A 140-kg horizontal beam is supported at each end. A 320-kg piano rests a quarter of the way from one end. What is the vertical force on each of the supports? 16, 19 Problem 2 Ch 9 #16: Three children are trying to balance on a seesaw, which consists of a fulcrum acting as a pivot at the center and a very light board 3.6 m long. Two children are already on either end. Boy A has a mass of 50 kg and girl B has a mass of 35 kg. Where should girl C, whose mass is 25 kg, place herself to balance the seesaw?

14 Problem 3 Ch 9 #19: A 172-cm-tall person lies on a light (massless) board which is supported by two scales, one under the top of her head and one beneath the bottom of her feet. The two scales read, respectively, 35.1 kg and 31.6 kg. What distance is the center of gravity of this person from the bottom of her feet? Problem Solving Deskwork AS/A2 page 82 all problems.

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